On the transient behaviour of stable linear systems

Size: px

Start display at page:

Download "On the transient behaviour of stable linear systems"

Chad Norton
5 years ago
Views:

1 On the transient behaviour of stable linear systems D. Hinrichsen Institut für Dynamische Systeme Universität Bremen D Bremen Germany A. J. Pritchard Mathematics Institute University of Warwick Coventry CV4 7AL England Keywords: Transient behaviour, uncertain systems, pseudospectra, spectral value sets. Abstract In this note we derive various estimates for the transient bound max t e At of a stable linear system. It is shown how this bound can be diminished by state feedback. Finally we analyze the robustness of an acceptable transient behaviour with respect to structured perturbations of A. 1 Introduction Trajectories of a stable linear system may temporarily move far away from the origin before approaching it as t. Such a transient behaviour is often exhibited by highly nonnormal systems. From a practical point of view, if the state excursions are very large the stable system actually behaves like an unstable one. Moreover, if the system is obtained by linearization of a nonlinear system around an equilibrium point the large transients of the linear part may incite the nonlinearities resulting in a complicated turbulentlike behaviour. In this case the practical instability of the equilibrium point is reflected by an extreme thinness of its domain of attraction. In fluid dynamics the interaction between large transient motions of the linear part and nonlinearities has recently been put forward as an explanation for experimental results where there are observed instabilities of flows at Reynolds numbers which differ significantly from those obtained via spectral analysis, see [1], [2], [3], [4]. It is a plausible conjecture that systems with large transient motions are somehow close to instability. By this we mean that some small perturbation may move some eigenvalues of A from the open left to the closed right half-plane. We will investigate this relationship in the present note by considering the union of all the spectra σ(a + ) where < δ. The set of all these spectral values of perturbed matrices has been called pseudospectrum, spectral portrait or spectral value set of A in the literature, see [5], [6], [7]. This concept has received growing attention in the nineties, see e.g. [8], [9],[1]. An excellent review of many results including some historical remarks and an interesting catalogue of examples is given in [11]. The organization of this paper is as follows: In Section 2 various estimates for the transient bound are obtained via the distance of A from the set of normal matrices and in terms of a differential Liapunov equation. In Section 3 we discuss the relationship between the transient bound and spectral value sets. In Section 4 it is shown how the transient bound can be reduced by state feedback. Finally, in Section 5 perturbation bounds are presented which ensure acceptable transient behaviour of a system with perturbed parameters. Proofs of most of the results in this paper can be found in [12]. 2 Transient Bounds Throughout this section it is assumed that A K n n, K = R or C and σ(a) C. Definition 2.1. The transient bound of the semigroup generated by A, (e At ) t is defined to be M (A) := max t eat, where is an operator norm. So if the maximum in the above expression is achieved at t and ˆx is such that e At ˆx = e At ˆx, then for every x K n, the solution x(t) of the initial value problem ẋ = Ax, x() = x satisfies x(t) M (A) x, t and for x = ˆx, we have x(t ) = M (A) ˆx. So M (A) is a measure of the relative (with respect to x ) maximum distance of the trajectory x(t), t from the origin. Clearly M (A) 1. If M (A) = 1, then (e At ) t is said to be a contraction semigroup. Since contraction semigroups have the best possible transient bounds it is interesting to determine necessary and sufficient conditions for A to generate such semigroups. These can be expressed in terms of the dissipativity of A with respect to any semi-scalar inner product on K n. Definition 2.2. Suppose is a norm on K n, then a semi-scalar product [, ] on K n is a scalar-valued function (x, y) [x, y] on K n K n such t hat for all x, y, z

2 K n, λ K [x + y, z] = [x, z] + [y, z], [λx, y] = λ[x, y], [x, x] = x 2, [x, y] x y. Definition 2.3. A matrix A K n n is dissipative with respect to a semi-scalar product [, ] if Re [Ax, x] for all x K n. The following theorem can be found in [13, IX.8]. Theorem 2.1. Let A K n n then A generates a contraction semi-group on K n if and only if A is dissipative (with respect to any semi-scalar inner product on K n ). As a corollary we obtain: Corollary 2.2. Suppose that K n has a Hilbert space structure and α R +, then if and only if e At e αt, t (1) A + A 2αI n, (2) where A is the adjoint of A with respect to the inner product in K n. Remark 2.3. Suppose thatk n is a Hilbert space with inner product, P 2, where P H n (Hermitian matrices), P and, 2 is the usual Euclidean inner product. Then the adjoint of A, A = P 1 A P where A is the complex conjugate and transpose of A. So A generates a contraction semigroup onk n, (with norm induced by the inner product, P 2) if and only if P A + A P. We see therefore that any matrix A with σ(a) C generates a contraction semigroup with respect to some norm induced by a particular inner product. The property of generating a contraction semigroup is robust. To see this let N C be the set of all n n matrices which do not generate a contraction semigroup. Proposition 2.4. Suppose that K n is a Hilbert space and A K n n generates a contraction semigroup. Then the distance dist(a, N C) of A from the set N C is given by dist(a, N C) = 1/2(λ max (A + A )). From now on it will be assumed that K n is provided with the usual inner product and the corresponding Euclidean norm. An important subset of matrices which generate contraction semigroups are the normal matrices. Let N denote the set of normal matrices in K n n. Lemma 2.5. Suppose A N and Then α(a) = sup{re λ; λ σ(a)}. (3) e At e α(a)t, t. (4) For every A K n n A, α(a) defined via (3) is called the spectral abscissa of A. This desirable property (4) is captured in the following definition. Definition 2.4. A matrix A is said to be stable with normal transient behaviour if for some α >, we have e At e αt, t. (5) Using the perturbation result below (an immediate consequence of Gronwall s Lemma) a condition for A to be stable with normal transient behaviour can be obtained. Lemma 2.6. Suppose that e At is a semigroup on K n satisfying e At Me αt, t (6) for some M 1 and α R. If K n n, then e (A+ )t Me (α M )t, t. (7) By Lemma 2.6 the following holds. Corollary 2.7. Suppose N N has spectral abscissa α(n). If A K n n satisfies δ = A N, then e At e (α(n) δ)t, t. (8) Since the set N of normal matrices is closed in K n n, for a given A, there exists an N N which is the closest to A: A N = dist(a, N ) =: δ(a). The set of all the normal operators N with this property is compact. By the continuity of the spectrum there exists a closest normal matrix of minimal spectral abscissa. Given any A we denote by N A one such closest normal matrix with minimal spectral abscissa (this matrix may not be uniquely determined). As a consequence of Corollary 2.7 we have Corollary 2.8. A matrix A K n n is stable with normal transient behaviour if α(n A ) > δ(a). (9) Another class of matrices for which there are well known estimates of the transient behaviour are the diagonalizable ones. Lemma 2.9. Suppose that A is diagonalizable and A = T DT 1 with diagonal D for some T Gl(n, C) then e At T T 1 e α(a)t, t. (1) For a general A and every α [, α(a)) let M α (A) be the smallest M satisfying e At Me αt, t. In particular, M (A) is equal to the transient bound of A. Now for every α [, α(a)) let t α be such that e At α = M α (A)e αt α

3 (such a t α exists). Then M α (A)e αtα M (A) = e At M α (A)e αt and hence M (A)e αt M α (A) M (A)e αt α. (11) As a final result of this section we relate M α (A) (and for α = the transient bound of a matrix A) to estimates for the solution of differential Liapunov inequalities. Proposition 2.1. Suppose α [, α(a)) and A α = A + αi n. Given an Hermitian matrix P H n, P such that P A αp P A α, P () = P (12) has a solution P α (t) satisfying P α (t) M 2 σ(p )I n, t, then M M α (A). For the special case P = I n, the solution of the equation in (12) satisfies sup t P α (t) = M α (A) 2. If for some Q H n, Q then P A αp P A α Q =, P () = P, P α (t) = e A α t P e A αt + t e A α (t s) Qe Aα (t s) ds. So the smallest bound for P α (t) is obtained for Q =. This suggests that we should have restricted our considerations to the equality in (12). We have chosen not to do so because this precludes the possibility of constant solutions. Indeed since σ(a α ) C, there exists a P, such that (12) has the constant solution P α (t) = P α = P. Then if P α we may choose M 2 σ(p α ) = P α = σ(p α ) and hence we obtain the bound σ(p α ) σ(p α ) 1 = P α P 1 α M α (A) 2. (13) So an interesting question is: How should one choose C K p n with (A, C) observable, such that for the unique solution P α of the Liapunov equation A αp + P A α + C C = the expression P α Pα 1 is minimized. If (2) holds, so that A + A + 2αI n, then we may choose P α = I n and hence obtain M α (A) = 1 as anticipated in Corollary Spectral Value Sets and Pseudo-Spectra In [7] we introduced unstructured spectral value sets. Definition 3.1. The unstructured spectral value set at level δ > is σ K (A, δ) = σ(a + ). K n n, <δ These spectral value sets are called real if K = R and complex if K = C. A characterisation of real spectral value sets is given in terms of the real µ-function, [8], but their computation is difficult. In contrast for the complex case we have [9]: σ C (A, δ) =σ(a) {s ρ(a); (si n A) 1 > δ 1 }. (14) where ρ(a) = C \ σ(a) denotes the resolvent set of A. This result connects spectral value sets with pseudo-spectra introduced by Trefethen [5]. Definition 3.2. The δ-pseudo-spectrum is Λ δ (A) := {s C; (si n A) 1 δ 1 } = σ C (A, δ), where by definition (si n A) 1 = if s σ(a). (15) Trefethen has made use of pseudo-spectra in order to obtain bounds on the transient behaviour. By using the equality e At = 1 e st (si A) 1 ds, 2πı Γ where Γ is a positively-orientated closed curve enclosing σ(a) in its interior, he obtains the following upper bound. Proposition 3.1. Also, using e At 1 2πδ (si A) 1 = Λ δ (A) e t Re s ds. e st e At dt, Re s >, he obtains the following lower estimate for the transient bound. Proposition 3.2. M (A) sup δ 1 α δ (A), α δ (A) = sup Re s. δ> s Λ δ (A) Remark 3.3. Although the problem of transient bounds has been posed on finite dimensional spaces many of the ideas and results can be extended to infinite dimensional ones. If α(a) = sup Re s, s σ(a) then for infinite dimensional spaces it is not necessarily the case that for any ω > α(a), there exists M ω such that e At M ω e ωt, t. However, it can be shown [11] that if α δ (A) = sup Re s s Λ δ (A) for any ω > lim δ α δ (A), there exists M ω such that e At M ωe ωt, t. In the previous section, in Corollary 2.8, the distance of a matrix from normality was used to get a sufficient condition for a matrix to be stable with normal transient growth. A similar condition in terms of spectral value sets can also be found.

4 Proposition 3.4. If δ(a) is the distance of A from normality. Then a sufficient condition for A to be stable with normal transient behaviour is σ C (A, δ(a)) C δ(a) (16) where C γ denotes, for every γ R, the open left half plane {s C; Re s < γ}. If σ(a) C δ(a) (otherwise (16) is clearly not valid) condition (16) is equivalent to σ C (A, δ(a)) ( δ(a) + ır) =. (17) It follows from (15) that (17) holds if and only if max (( δ(a) + ıω)i n A) 1 < δ(a) 1. (18) ω R Now it is clear that (σ C (A, δ)) C δ for small δ >. If this is not the case for some δ (e.g. δ = α(a)) then it will not be satisfied for any δ δ. Hence there exists exactly one δ > such that Lemma 4.1. Suppose R H n, R and consider the differential Liapunov equation Ż AZ ZA + BRB =, t. (2) The unique solution of (2) on R + with initial value Z() = I n is given by Z(t) = e At [I n t ] e As BRB e A s ds e A t. (21) Let I + = [, t + ) = {t ; Z(t) }, then < t + and the initial value problem Ẋ + A X + XA XBRB X =, X() = I n. (22) has the solution P (t) = Z(t) 1 on I +. Moreover lim t t+ P (t) =. If F (t) = RB P (t), t I +, then ϕ F (t, )ϕ F (t, ) Z(t), t I +. (23) We see from (21) and (23) that ϕ F (t, ) e At, t I + and if the pair (A, B) is controllable the inequality is strict for t (, t + ). This suggests a possible way of reducing the transient bound M (A) to a more acceptable level M. and (σ C (A, δ)) C δ, δ < δ (19) (σ C (A, δ )) C δ, δ δ. Proposition 4.2. Suppose that M 1 is an acceptable transient bound and there exists a T (, t + ) such that Z(T ) M 2 /M (A) 2, Z(t) M 2, t [, T ] (24) This δ is uniquely determined by the equation δ 1 = max ω R (( δ + ıω)i n A) 1. So the condition for A to be stable with normal transient behaviour given in Proposition 3.4 is δ(a) < δ. This seemingly nice result suffers because although there is a computational scheme for determining δ(a) with respect to the Frobenius norm [14], to our knowledge there does not seem to be one for the spectral norm. 4 State Feedback In this section the possibility of improving the transient bound by stabilizing state feedback is considered. Suppose B K n m, F : [, ) K m n is piecewise continuous and the controlled dynamics are ẋ = (A + BF (t))x, x() = x K n. If ϕ F (, ) is the evolution operator generated by A + BF (t) its transient bound is defined to be M (A + BF ( )) = max t ϕ F (t, ). We have the following lemma. where Z( ) is as in Lemma 4.1. Then for { RB F (t) = P (t) t [, T ] t > T where P ( ) is as in Lemma 4.1 one has M (A + F ( )) M. In order to check whether or not the conditions in the above proposition hold, one needs to compute the solution of the differential Liapunov equation (2) with initial value Z() = I n and monitor whether or not it is possible to find T such that Z(T ) M 2 /M (A) 2 and on the interval [, T ], Z(t) does not transcend the value M 2 and σ(z(t)) >. In order to get more insight into this let C R (t) be the controllability Gramian of ( A, BR 1/2 ) on the interval [, t], then from (21) we see that Z(t) provided C R (t) = t In particular we have e As BRB e A s ds I n. (25) t + = sup{t R + ; det(i n C R (s)), s [, t]}. Now assume that (A, B) is controllable, then T C R (T ) σ(r) e As BB e A s ds.

5 For any R such that (25) holds with t = T, let β (R) = σ(c R (T ))/σ(c R (T )), then (1 β (R)) 1/2 is a measure of the eccentricity of the ellipsoid described by {x K n ; x, C R (T )x = 1}. Now β (R) = β (αr), α > and provided α < σ(c R (T )) 1 then (25) will continue to hold at t = T when R is replaced by αr. In which case from (21) one has Z(T ) (1 ασ(c R (T ))) e AT 2. The choice α = σ(c R (T )) 1 yields Z(T ) (1 β (R)) e AT 2 So just as in (13) in order to obtain a good bound one would try to choose R to make the eccentricity small, i.e. β (R) close to one. We will now give an interpretation of the solution P (t) of the initial value problem (22) in terms of the following finite time optimal control problem: Minimize T J(x, u) = x(t ), P (T )x(t ) + u(t), R 1 u(t) dt subject to ẋ = Ax + Bu, x() = x. Let F (t) = RB P (t), t [, T ] then we have for every control u( ) L 2 (, T ; K m ), d x(t), P (t)x(t) dt = x(t), ( d dt P (t) + A P (t) + P (t)a)x(t) +2 Re x(t), P (t)bu(t) = u(t) F (t)x(t), R 1 (u(t) F (t)x(t)) u(t), R 1 u(t), t [, T ]. Hence integrating from to T, yields J(x, u) = x, P ()x + T u(t)+f (t)x(t), R 1 (u(t)+f (t)x(t)) dt. So the control u(t) = RB P (t)x(t) minimizes the cost functional J(x, u) and the optimal cost is x 2. Note that this is a characterization a posteriori since the cost functional J(x, u) depends on the final value P (T ). 5 Robustness of acceptable transients Consider the uncertain system ẋ = A x := (A + D E)x where (A, D, E) K n n K n l K q n. Assume that the nominal system ẋ = Ax is Hurwitz stable and has an acceptable transient behaviour expressed by the condition M (A) = max t eat < M where M > 1 is a given bound. We now introduce a measure for the robustness of this condition under perturbations A A. Definition 5.1. Given an acceptable level M > 1, then the radius of acceptable transient behaviour of A under perturbations of the form A A + D E, K l q is defined by r K (A; D, E; M) = inf { ; K l q, t : e (A+D E)t M}. Proposition 5.1. Suppose there exists P H n, Q H q, R H l, P, Q, R such that P A P P A E QE P DRD P =, has a solution on R + which satisfies P (t) M 2 σ(p ) I n, t. Then r K (A; D, E; M) (σ(q) σ(r)) 1/2. P () = P (26) Example 5.2. Suppose A is a normal matrix, A = U diag (λ 1, λ 2,..., λ n ) U, with U unitary, P = I, Q = α 2 I q, R = β 2 I l and and D = E = I, then if we set ˆP = UP U = diag (p 1 (t), p 2 (t),..., p n(t)), (26) is equivalent to the following set of n decoupled scalar differential Riccati equations: p i (λ i +λ i )p i α 2 β 2 pi 2 =, p i () = 1, i n. Let γ i = (λ i + λ i )/2, γ 1 γ 2... γ n and choose α 2 = β 2 = γ 1, then p 1 (t) 1 and p i(t) 1 for all t and i n. So 1 ˆP (t) = P (t), t. Thus rk(a; I, I; 1) γ 1 = 1/2(λ max(a + A )), a result already anticipated in Proposition 2.4. Finally consider the nonlinear equation ẋ = Ax + D (Ex), x() = x, (27) where : K q K l is locally Lipschitz and satisfies () =. The following proposition extends Proposition 5.1 to nonlinear perturbations of the form A A where A (x) = Ax + D (Ex). Proposition 5.3. Under the assumptions of Proposition 5.1 suppose that (z) (σ(q)σ(r)) 1/2 z, z K q. Then for every x K n, there exists a unique solution x(, x ) of (27) on R + and x(t, x ) M x, for all t.

6 References [1] Gustavsson, L. H., Energy growth of three dimensional disturbances in plane, Poiseuille flow, J. Fluid Mech., Vol. 224, pp , [2] Butler, K. M. and Farrell, B. F., Three dimensional optimal perturbations in viscous shear flow, Phys. Fluids A Vol. 4, pp , [3] Trefethen, L. N., Trefethen, A. E., Reddy, S. C., and Driscoll, T. A., Hydrodynamic stability without eigenvalues, Science, Vol. 261, pp , [4] Reddy, S. C., Schmidt, P., and Henningson, D., Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math., Vol. 53, pp , [5] Trefethen, L. N., Approximation theory and linear algebra, Algorithms for Approximation II, Chapmah and Hall, 199. [6] Godunov, S. K., Spectral portraits of matrices and criteria of spectrum dichotomy, Proc. of Internat. Conf. on Computer Arithmetic, Scientific Computation and Mathematical Modelling. SCAN-91, Oldenburg, [7] Hinrichsen, D. and Pritchard, A. J., On the robustness of stable discrete time linear systems, New Trends in Systems Theory, Proc. Conf. Genova 199, pp , Birkhäuser [8] Hinrichsen, D. and Kelb, B., Stability radii and spectral value sets for real matrix perturbations, in U. Helmke and R. Mennicken, editors, Proc. Conf. MTNS, Regensburg, [9] Hinrichsen, D. and Kelb, B., Spectral value sets: a graphical tool for robustness analysis, Systems & Control Letters, Vol. 21, pp , [1] Boettcher, A., Pseudospectra and singular values of large convolutions operators. J. Integral Eqs. and Appl., Vol. 6, pp , [11] Trefethen, L. N., Pseudospectra of linear operators, SIAM Rev., Vol. 39, pp , [12] Pritchard A. J., Transitory behaviour of uncertain systems. In F. Colonius et al. (eds.), Advances in Mathematical Systems Theory, pp. 1-18, Birkhäuser, Boston, 2. [13] Yosida, K., Functional Analysis., Springer-Verlag, Berlin-Heidelberg-New York, [14] Ruhe A., Closest normal matrix finally found! BIT, Vol. 27, pp , 1987.

The goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T

1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.